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When solving logarithmic equations, converting to exponential form is a critical step. Understanding the relationship between logarithms and exponents is essential. To go from logarithmic to exponential form, remember the rule: \( \log_b{a} = c \) means \( b^c = a \). This conversion helps you eliminate the logarithm and deal directly with the numbers.

For the equation \( \log{(x + 100)} = 3 \), we use the common logarithm base, which is 10. So, \( \log{(x + 100)} = 3 \) translates to \( 10^3 = x + 100 \). This simplifies the problem significantly.

When you rewrite logarithmic equations in their exponential form:

• You make it easier to isolate the variable.
• It becomes straightforward to solve using arithmetic operations.

Remember, mastering this conversion can make logarithmic equations much simpler.

The common logarithm is a logarithm with base 10. It is represented as \( \log{x} \) without a base. It is commonly used in scientific calculations and standard mathematical problems.

In the equation \( \log{(x + 100)} = 3 \), the absence of a base means we are dealing with a common logarithm, or base 10. This makes calculations simpler because the base is consistent and well-understood.

The properties of common logarithms include:

• Easy conversion to exponential form since \( \log_b{x} = y \) converts to \( 10^y = x \).
• Frequently used in practical applications, from measuring the intensity of earthquakes to calculating compound interest rates.

Understanding common logarithms is crucial for both academic exercises and real-life mathematical applications.

Solving logarithmic equations involves a series of steps that simplify the problem. Let's break down the process discussed in the solution:

First, write down the equation: \( \log{(x + 100)} = 3 \). By converting this to exponential form, you get \( 10^3 = x + 100 \). This removes the logarithm and simplifies the equation.

Next, you calculate \( 10^3 = 1000 \), leaving you with \( 1000 = x + 100 \).

Then, solve for \( x \) by isolating it on one side. Subtract 100 from both sides of the equation: \( 1000 - 100 = x \). You then get \( x = 900 \).

Finally, verify the solution by substituting \( x \) back into the original equation: \( \log{(900 + 100)} = \log{1000} = 3 \). This confirms the correctness of \( x \).

Following these steps helps you tackle logarithmic equations systematically:

• Convert to exponential form.
• Simplify and solve for the variable.
• Verify the solution to ensure accuracy.

With practice, you'll become proficient at solving even complex logarithmic equations.